The goal of characterizing such nerves, without an a priori definition of “weak -category” to start from, is called simplicial weak ω-category theory. It is expected that the (nerves of) weak -categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.” General weak complicial sets can be regarded as “presentations” of weak -categories.
Weak complicial sets are a joint generalization of
be the stratified simplicial set whose underlying simplicial set is the -horn of , with marked cells those that are marked in ;
be obtained from by making the st -face and the st face thin;
be obtained from by making all -faces thin.
An elementary anodyne extension in , the category stratified simplicial sets is
for and .
A complicial set is a weak complicial set in which such liftings are unique.
For a strict ω-category and its ω-nerve, the Roberts stratification which regards each identity morphism as a thin cell makes a strict complicial set, hence a weak complicial set. This example is not “saturated.”
There is also the stratification of which regards each -equivalence morphism as a thin cell. with this stratification is a weak complicial set (example 17 of Ver06). This should be the “saturation” of the previous example, and exhibits the inclusion of strict -categories into weak ones.
A simplicial set is a weak complicial set when equipped with its maximal stratification (every simplex of dimension is thin) if and only if it is a Kan complex. This example is, of course, saturated, and is viewed as embedding -groupoids into -categories.
A simplicial set is a quasi-category if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its homotopy category). It presents the embedding of -categories into weak -categories.
Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why can “correctly” be regarded as a full subcategory of . This is not true at higher levels; for instance not every simplicial map between nerves of strict -categories necessarily preserves -equivalence morphisms.
The definition of weak complicial sets is definition 14, page 9 of
Further developments are in